Riemann流形上Laplace算子第一特征值的一点注记

本文研究了黎曼流形上Laplace算子的第一特征值,利用流形的测地球上的Sobolev常数进行讨论并进行Moser迭代,得到闭的黎曼流形上Laplace算子第一特征值的一个下界估计.     

加权Laplace在积分Ricci曲率条件下的特征值估计

本文研究了积分Ricci曲率条件下加权Laplace算子的第一特征值估计的问题.利用Bochner公式与加权Reilly公式等处理特征值问题的方法,获得了加权Laplace在积分Ricci曲率条件下第一特征值估计下界的估计.     

Wilson元特征值下逼近准确特征值

该文讨论矩形域上Laplace算子特征值问题有限元近似.证明了Wilson非协调有限元特征值下逼近准确特征值,从而解决了有限元法中长期存在的一个猜想.     

Carnot群上水平Laplace算子的二次多项式算子的特征值不等式（英文）

本文研究了Carnot群上水平Laplace算子的二次多项式算子的Dirichlet特征值问题,并建立了一些特征值不等式.特别地,我们的结果涵盖了文献[10]对双调和水平Laplace算子所获得的结果.     

欧氏空间子流形的第一特征值的估计

本文利用浸入在欧氏空间中的子流形的第二基本形式的长度平方估计其Laplace算子的第一特征值的上界,从而建立紧致子流形等距同构于球面的一个特征.     

Yamabe流上Laplace算子特征值的一个单调性应用

讨论了紧致无边流形上Laplace算子的特征值在Yamabe流上随时间的变化情况,结合极值原理得到了Laplace算子特征值的单调性.     

图的特征值交错方法的几个应用

本文利用特征值交错方法研究了图的谱半径下界等问题,得到了图谱半径的两个新的紧下界,以及图的Laplace谱与四边形个数的一个关系式.     

具有加权测度的H型群上漂移Laplace算子的Levitin-Parnovski型特征值不等式

本文研究了具有加权测度dμ=e~(-φ)dv的H型群G上漂移Laplace算子-?_G+▽_Gφ,▽_G(·)的Dirichlet特征值问题,建立了该问题的Levitin-Parnovski型特征值不等式,推广包含了Ilias和Makhoul对Heisenberg群上次Laplace算子所获得的结果 (J. Geom. Anal.,2012, 22(1):206–222).     

偏微分方程特征值问题的弱有限元方法

本文简要回顾弱有限元方法在偏微分方程特征值问题中的应用;对于一般椭圆型特征值问题,给出弱有限元方法的分析框架,并以Laplace特征值问题为例给出理论分析.本文对特征值问题的弱有限元方法研究进展进行综述,展望今后准备开展的工作.     

小负曲率流形上Laplace算子第一特征值的下界估计

本文首先对流形的测地球上的Sobolev常数进行讨论,并利用它进行Moser迭代,最终得到具有小负曲率的闭的黎曼流形上Laplace算子特征值的一个下界估计.     

On the Laplacian integral tricyclic graphs

A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.     

A characterization of spectral integral variation in two places for Laplacian matrices

We describe the graphs having the property that adding a particular edge will result in exactly two Laplacian eigenvalues increasing by 1 and the other Laplacian eigenvalues remaining fixed. For a certain subclass of graphs, we also characterize the Laplacian integral graphs with that property. Finally, we investigate a situation in which the algebraic connectivity is one of the eigenvalues that increases by 1.     

Laplacian spectral characterization of 3-rose graphs

A 3-rose graph is a graph consisting of three cycles intersecting in a common vertex, J. Wang et al. showed all 3-rose graphs with at least one triangle are determined by their Laplacian spectra. In this paper, we complete the above Laplacian spectral characterization and prove that all 3-rose graphs are determined by their Laplacian spectra.     

On the equivalence of heat kernel estimates and logarithmic Sobolev inequalities for the Hodge Laplacian

In this paper we consider the Hodge Laplacian on differential k-forms over smooth open manifolds MN, not necessarily compact. We find sufficient conditions under which the existence of a family of logarithmic Sobolev inequalities for the Hodge Laplacian is equivalent to the ultracontractivity of its heat operator.We will also show how to obtain a logarithmic Sobolev inequality for the Hodge Laplacian when there exists one for the Laplacian on functions. In the particular case of Ricci curvature bounded below, we use the Gaussian type bound for the heat kernel of the Laplacian on functions in order to obtain a similar Gaussian type bound for the heat kernel of the Hodge Laplacian. This is done via logarithmic Sobolev inequalities and under the additional assumption that the volume of balls of radius one is uniformly bounded below.     

Lévy Laplacians in Hida Calculus and Malliavin Calculus

Some connections between different definitions of Lévy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev–Schwartz distributions over the Wiener measure (Hida calculus). One can consider the chain of Lévy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Lévy Laplacian. Another approach to defining the Lévy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (Malliavin calculus). It is proved that the Lévy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Lévy Laplacians, but not with the classical Lévy Laplacian, under the embedding of the Sobolev space over the Wiener measure in the space of generalized functionals over this measure. It is shown which Lévy Laplacian in the stochastic analysis is connected with the gauge fields.     

Gross Laplacian Acting on Operators

In this paper, we introduce a noncommutative extension of the Gross Laplacian, called quantum Gross Laplacian, acting on some analytical operators. For this purpose, we use a characterization theorem between this class of operators and their symbols. Applying the quantum Gross Laplacian to the particular case where the operator is the multiplication one, we establishes a relation between the classical and the quantum Gross Laplacians.      

Constructably Laplacian integral graphs

A graph is Laplacian integral if the spectrum of its Laplacian matrix consists entirely of integers. We consider the class of constructably Laplacian integral graphs - those graphs that be constructed from an empty graph by adding a sequence of edges in such a way that each time a new edge is added, the resulting graph is Laplacian integral. We characterize the constructably Laplacian integral graphs in terms of certain forbidden vertex-induced subgraphs, and consider the number of nonisomorphic Laplacian integral graphs that can be constructed by adding a suitable edge to a constructably Laplacian integral graph. We also discuss the eigenvalues of constructably Laplacian integral graphs, and identify families of isospectral nonisomorphic graphs within the class.     

The least eigenvalue of signless Laplacian of graphs under perturbation

We investigate how the least eigenvalue of the signless Laplacian of a graph changes by relocating a bipartite branch from one vertex to another vertex, and minimize the least eigenvalue of the signless Laplacian among the class of connected graphs with fixed order which contains a given non-bipartite graph as an induced subgraph.     

On the second largest Laplacian eigenvalues of graphs

The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.     

Sharp Stability of Some Spectral Inequalities

In this work we review two classical isoperimetric inequalities involving eigenvalues of the Laplacian, both with Dirichlet and Neumann boundary conditions. The first one is classically attributed to Krahn and P. Szego and asserts that among sets of given measure, the disjoint union of two balls with the same radius minimizes the second eigenvalue of the Dirichlet–Laplacian, while the second one is due to G. Szegő and Weinberger and deals with the maximization of the first non-trivial eigenvalue of the Neumann–Laplacian. New stability estimates are provided for both of them.